Valve seat with trapped O-ring

ABSTRACT

A valve seat (28) having a generally saddle-shaped, eccentric surface (52); a groove (50) of constant depth and width formed in the eccentric surface, having sidewalls (70) perpendicular to the eccentric surface and a base (72) parallel to the eccentric surface; and a flexible O-ring (54) retained by friction within the groove. The groove sidewalls and base have a surface finish in the range of about 32 to 125, which provides sufficient smoothness to retain the O-ring in the required non-planar shape. 
     The valve seat groove (50) can be produced according to the disclosed method, which is preferably implemented in a numerically controlled four-axis horizontal milling tool system. The valve seat part (100) is mounted in a fixture (102) on a rotary table (104) such that the part axis (56) is aligned with the longitudinal axis (Z) of the tool (106). Once the initial penetration position and depth of the tool (106) is established on the eccentric surface (52a) of the part (100), the only required machine movements are table rotation (104) and tool vertical axis movement (Y).

This is a division of application Ser. No. 444,565 filed Nov. 26, 1982,now Pat. No. 4,492,392.

BACKGROUND OF THE INVENTION

This invention relates to valve components having a resilient sealingsurface, and in particular to a valve seat having a groove or step formounting a resilient seal.

Conventional gate valves typically include a valve body, a valve seatassembly contained within the valve body, and an actuating assembly formoving the seat assembly to control the fluid flow through the body.Commonly, the valve seat is a generally cylindrical hollow componenthaving a surface at one end adapted to mate with a portion of the valvebody. Since the seat and valve body are typically perpendicularlyoriented cylinders, their "intersection" is not planar, but rathereccentric or saddle-shaped. Conventionally, a resilient material ismounted onto the saddle-shaped surface of the valve seat, to assure atight seal against the valve body. One known technique is to machine agroove or step in the valve seat surface, then mold vulcanized rubber ofapproximately 90 durometer to the groove or step.

The effectiveness and longevity of the valve are usually limited by theprecision and durability of the valve seat seal. In time, the rubberdeteriorates and the valve must be repaired. Conventionally, suchrepairs require replacing the entire valve seat, since it is impossibleto replace, in the field, only the molded rubber. As is well known,proper molding of the rubber onto the metal seat requires carefulsurface preparation and controlled time and temperature conditions whichare not normally available where gate valves are frequently used, e.g.,oil and gas wells.

The limitations of presently known techniques for fabricating valveseats having resilient seals, and the inability to quickly repairdeteriorated seals, have resulted in high costs for manufacturing andmaintaining gate valves for use in oil and gas fields. Accordingly,there exists a great and long felt need for improvements to gate valveseats that would reduce the cost of manufacture and enable users toquickly and inexpensively replace worn resilient seals.

SUMMARY OF THE INVENTION

The present invention is desired to a novel valve seat havingparticularly advantageous use in gate valves of the type generally usedin oil field exploration and production. The invention also includes anovel method of manufacturing the valve seat.

According to the invention, the valve seat has a generallysaddle-shaped, eccentric surface; a groove of constant depth and widthformed in the eccentric surface, having sidewalls perpendicular to theeccentric surface and a base parallel to the eccentric surface; and areplaceable, flexible O-ring retained by friction within the groove. Thegroove sidewalls and base have a surface finish in the range of about 32to 125, which provides sufficient smoothness to retain the O-ring in therequired non-planar shape, yet is rough enough to be within theproduction capabilities of numerically controlled machine tools adaptedto operate according to the inventive method.

According to the inventive method, a novel machine tool fixture and toolmotion sequence enables a conventional four axis horizontal milling toolsystem to automatically cut the required O-ring groove perpendicular tothe eccentric surface, with a groove finish in the range of about32-125.

The method is preferably utilized in a numerically controlled system,and preferably includes the steps of arranging a four axis machine suchthat the axis of rotation B of the rotary work table is vertical, the Xaxis of the tool is horizontal, the Y axis of the tool is vertical, andthe Z axis of the tool is mutually perpendicular to the X and Y axesalong a line perpendicularly intersecting the B axis. The generallycylindrical valve seat part having an eccentric surface is attached tothe rotary table fixture such that the axis of the seat is coincidentwith the Z axis of the tool. An end mill cutting tool is positionedalong the X axis until the tool contacts the valve seat eccentricsurface. The tool has a cutting width slightly larger than that, of theundeformed O-ring. The tool motion along the Z axis is set for aconstant depth of cut slightly less than the undeformed O-ring width. Ageometrical relationship is established between the desired radius R_(G)of the O-ring groove, for controlling the Y axis tool movement, and theradius R_(P) of the eccentric surface of the valve seat part, fordetermining the initial rotation around the B axis to align the toolperpendicularly to the eccentric surface, and to control the Y axisincremental movement accompanying each B axis incremental rotation.Finally, the machine is provided with control instructions or programthat represents the desired groove path by a series of incrementalposition coordinates requiring tool movement only along the Y axis incooperation with the seat part rotation about the B axis.

The invention thus provides a valve seat having a saddle shaped grooveinto which a flexible O-ring can be manually installed by merelydeforming and compressing the ring into the groove. The ring compressionagainst the groove walls and base captures or traps both lateral sidesof the ring without additional adhesives required.

A deteriorated ring can be easily removed from the seat without damagingthe eccentric surface or groove. A new ring can be immediately installedin the groove, and it is sufficiently trapped in the groove to enablethe maintanence operative to return the valve seat to the valve seatassembly without fear that the ring will "pop" out of the groove as theentire valve is reassembled.

The present invention reduces the cost of manufacturing the seat, sincethe rubber molding and vulcanizing is not required, and greatlysimplifies the maintenance of valves in the field.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic exploded view of the major parts of a knownhydraulic gate valve.

FIG. 2 is a perspective view of a valve seat according to the invention,adapted to replace the conventional cylindrical valve seat shown in FIG.1, with a clockface reference system superimposed thereon.

FIG. 3 is composite profile view of the valve seat shown in FIG. 2,where the portion at 12 o'clock is shown in section and the portionbetween 6 o'clock and 9 o'clock is shown with the resilient O-ringomitted for clarity.

FIG. 4 is an enlarged detail view of a portion of FIG. 3, showing thegroove at the 12 o'clock position.

FIG. 5 is a section view taken along line 5--5 of FIG. 3, showing thegroove perpendicularly oriented to the eccentric surface at the 9o'clock position.

FIG. 6 is a schematic view of a groove cut into the eccentric surface,illustrating, in exaggeration, an imperfect groove finish.

FIG. 7 is a perspective view of the valve seat part prior to the firststep of the inventive method, showing the eccentric surface previouslymachined thereon.

FIG. 8 is a schematic view of the valve seat part shown in FIG. 7,clamped into a fixture on a rotary table of a numerically controlledfour axis horizontal milling tool system, with the table axis B and toolaxes X,Y,Z, identified.

FIG. 9 is a diagram showing the typical relationship between circlesdefined by radii R_(P) of the part eccentric surface and R_(G) of theO-ring groove, whereby the Y axis tool movement can be determined for agiven fixture rotation about the B axis.

FIG. 10 is a schematic view showing the first penetration of the toolinto the eccentric surface at the 3 o'clock position.

FIG. 11 is a schematic view showing the continued milling by the toolthrough the seat surface at the 6 o'clock position.

FIG. 12 is a schematic view showing the continued milling by the toolthrough the seat surface at the 9 o'clock position.

DESCRIPTION OF THE PREFERRED EMBODIMENT

FIG. 1 shows, in schematic rendition, the major functional parts of awell-known gate valve 10 of the type used, for example, in oil fieldexploration and production. The valve includes a body 12 defining a mainbore 16, inlet and outlet nozzle 18, and flanges 20 for connecting thevalve body to inlet and outlet piping (not shown). The nozzles 18 areeither aligned or blocked, to permit or shut off flow, respectively, bythe position of the valve seat assembly 22 within the main bore 16. Asshown in an exploded view above the main bore 16, the seat assembly 22comprises outer 24 and inner 26 gates, valve seats 28a, and valve seatcarriers 30. The valve seat assembly 22 is positioned by the operatorbody 34, valve stem 36, and automatic or manual actuator means, such aswheel 38.

The present invention is directed to improvements in the design andmachining of the valve seat 28a. Although conventional seats 28a includea resilient seal 40 molded or otherwise affixed to the seat, sealreplacement or repair has been particularly costly and inefficient.

VALVE SEAT DESIGN

As shown in FIG. 2, the present invention provides a valve seat 28having a machined, functional groove 50 in an oval, eccentric valve seatsurface 52. The eccentric surface is saddle-shaped, since it mustconform to the cylindrical curvature of the main bore hole 16 of thevalve body (see FIG. 1). Essentially, the valve seat 28 and valve body16 can be represented by the perpendicular intersection of two cylindershaving different diameters. The purpose of the groove 50 is to provide ameans for sealing the seat 28 with a retained elastomeric seal ring, or"O" ring 54 (shown in section before installation within the groove).The valve seat improvements disclosed herein can readily be incorporatedinto present valve designs, and, for example, a conventional seat 28acould be removed from an existing gate valve of the type shown in FIG.1, and replaced with the improved seat 28 as shown in FIG. 2. Subsequentmaintenance would require replacement of only the O-ring seal 54; theseat 28 could be reinserted with a new seal 54.

In FIG. 2, the improved seat 28 is shown oriented consistently with thedepiction of the conventional valve seal 28a shown in FIG. 1. Forconvenience in understanding the ensuing description, the positions ofpoints of interest on the eccentric surface 52 will be referred to byclock position, e.g., 12 o'clock and 6 o'clock are shown in FIG. 2 asthe top and bottom positions, respectively. It may be appreciated that,if a vector were located normal to the eccentric surface 52 at the 12o'clock position, it would be parallel to the seat axis 56. If the baseof the vector were to travel along the eccentric surface, whileremaining perpendicular to it, the vector would increasingly pointoutwardly away from the seat axis, reach a maximum outward direction atthe 3 and 9 o'clock positions, then return to a parallel orientation tothe seat axis at the 6 o'clock position.

According to the invention, the groove 50 is always perpendicularlyoriented to the eccentric surface 52. Such groove orientation, incombination with the groove finish to be discussed more fully below,enables the O-ring 54 to be "trapped" in the groove, despite theconsiderable out-of-plane distortion required for the O-ring to conformto the groove contour.

Referring now to FIGS. 3, 4 and 5 additional details of the seat 28 willbe set forth. The seat is oriented such that a section view is shownthrough the 12 o'clock position, where the eccentric surface 52 andgroove 50 are parallel to the seat axis 56, and a profile view is shownbetween the 6 o'clock and 9 o'clock positions. The seat typically has acollar 58, a cylindrical neck portion 60 of uniform inner and outerdiameters, and the saddle-shaped eccentric surface 52. The eccentricsurface and saddle-shape are uniquely defined by the neck thickness 62and the part radius R_(P), which is equal to the nominal radius of thevalve body main bore 16.

The eccentric surface 52 has a generally rectangular groove 50 machinedtherein, leaving an eccentric rim 63 and a secondary seal surface 64,which is a backup to the primary, resilient seal provided by the O-ring54. In the preferred embodiment of the invention, the gate valve size(seat I.D.) is between about two to six inches. In a four inch valve,the groove 50 preferably has a depth D of 0.114 inch and a width W of0.156 inch, to accommodate a 90 durometer viton O-ring having anundeformed diameter of 0.139 inch. Thus, the O-ring would protrude 0.025inch above the eccentric surface 52, yet by self-retained or trappedtherein. The groove radius R_(G) is also shown on FIG. 3. A line can bedrawn along the base of the groove 72, that is always a perpendiculardistance R_(G) from the seat axis 56.

The appropriately sized O-ring 54 is chosen by determining the sealdiameter of the flat surface of the seat, then selecting an O-ring tofit the same approximate seal diameter. The O-ring is then stretched tofit the back diameter of the seat.

As indicated above, the automatic trapping of the O-ring 54, which istypically in the range of about 80-90 durometer, is strongly dependenton the machined finish within the groove 50. FIG. 4 shows the grooveinner and outer sidewalls 70 and base 72. As a practical matter, themachining of a groove such as 50 is a complex and painstaking procedure,and was not readily achievable on a production basis prior to theinventive method set forth below. Nevertheless, even with the inventivemethod, a preferably smooth groove cannot be achieved. We havediscovered, however, that perfection is not necessary; the O-ring can betrapped if the groove sidewall and base surfaces 70,72 are machined witha finish in the range of 32 to 125, with about 63 being quitesatisfactory. As illustrated in FIG. 6 for purposes of the presentdisclosure, groove finish is a measure of the piece-wise linearitybetween adjacent "ridges" or corners 66,68 resulting from the cuttingaction of the tool, measured along the groove or O-ring path. In cuttingthe groove, the machine tool cuts a sequence of arcs or chords; the sizeor span of each arc is related to how closely the groove base orsidewall approaches perfection. A 32 finish means the arithmetic averageof the ridge heights from the mean height, is 32 microinches.

It should also be understood that, although the sidewals 70 are parallelto each other and perpendicular to the eccentric surface 52, a deviationin perpendicularity of up to about 21/2° can be tolerated.

METHOD OF MACHINING THE GROOVE

The novel valve seat having a resilient seal trapped within a grooveperpendicularly oriented relative to an eccentric, saddle-shapedsurface, can be fabricated according to the following novel method forcutting such a groove.

FIG. 7 shows the valve seat part 100 as a conventionally machined blank,ready for further machining into a valve seat. The eccentric,saddle-shaped surface 52a is adapted to conform to the inner surface ofthe valve main bore 16 (see FIG. 1).

FIG. 8 shows the first step in the method, which consists of arranging afour-axis numerically controlled (N.C.) cutting tool system such thatthe fixture 102 is vertically oriented along the axis of rotation B ofthe rotary table 104, and the X (horizontal), Y (vertical), and Z(longitudinal) movement axes of the tool 106 are oriented mutuallyperpendicularly. Of course, other symbols or sign conventions could beused, depending on the particular N.C. machine being employed.

The fixture 102 includes means for fixedly mounting the seat part 100perpendicularly to the B axis of rotation, along the Z axis of tooltravel. A serrated expander collet 108 is shown in the figure, but othermeans could be used. It is important that the perpendicular distancefrom any point on the eccentric surface 52a of the seat part 100 to theB axis, be identical to the part radius R_(P). Another way of describingthis is that the B axis must be at the center of the radius of curvatureof the eccentric surface 52a.

Having established the coordinate system with the tool origin of theaxis 56 of the seat part, the N.C. machine must be controlled orprogrammed so the tool will traverse a path that will cut or mill thedesired groove in the eccentric surface. As will be appreciated as thedescription proceeds, once the tool is positioned along the X axis andonce the depth of cut is set on the Z axis, the tool follows a pathrequiring motion only along the Y axis, while the seat part turns byrotation about the B axis.

For convenience, it is desired that the tool initially penetrate theeccentric surface 52a at the 3 o'clock position, as shown in FIG. 10,and it is further desired that the tool penetrate the eccentric surfaceperpendicularly. The relationship between the initial Y penetrationpoint, Y₀, and the initial rotary table angle A₀, as well as theincremental relationship, i.e., the Y axis movement required for eachincrement in B axis rotation, must be determined.

Referring now to FIG. 9, the geometry of the system is shown forillustrating the derivation of the tool movement control functions inputto the N.C. machine. FIG. 9 shows circle 1 (in plane X-Y), drawn withthe radius R_(G) of the groove from the centerline of the seat part, andcircle 2 (in plane X-Z) having a radius R_(P). Starting at the 3 o'clockposition at radius R_(G) of circle 1, line 1 is extended upwardly frompoint 11 to intersect line 21 of circle 2. (In the figure, lines andpoints on or within circle 1 are designated LN 10, LN 11, LN 12 . . . ,and PT 10, PT 11 . . . ; lines and points on or within circle 2 aredesignated LN 20, LN 21 . . . , and PT 20 . . . , respectively; andlines connecting the circles are designated LN 1, LN 2 . . . ). Thusdrawn, line 1 defines points 21 and 22 and lines 21 and 22. The lengthof line 22 is equal to R_(P), and the length of line 21 is equal toR_(G). Therefore, the initial table angle A₀ is determined through therelationship,

    cos A.sub.0 =line 21/line 22

or

    cos A.sub.0 =R.sub.G /R.sub.P

Having established the initial position of the tool at Y₀ =0 (PT 11) andthe table at A₀ =arc cos R_(G) /R_(P), the path required to generate theentire groove must also be specified. As previously indicated, the toolmotion will be only vertical, and the table will rotate correspondingly,or vice versa. If it is assumed that for the first cut path step thework table angle changes from A₀ (the initial position of toolpenetration) to A₁, as shown on circle 2, then the necessary Y axis toolmovement will be that represented by the difference in Y axis valuebetween point 11 and point 12 on circle 1. Point 12 is determined bydropping line 2 from point 23.

As shown on circle 1, an isosceles triangle may be drawn by connectingpoints 10,11, and 12, with phantom line 11 drawn as the bisector. Theportion of line 11 indicated by T is the tolerance of the groove from atrue circle, i.e., T indicates the maximum deviation of the chordsegment in plane X-Y, represented by line 12, from the true arc joiningpoints 11 and 12 when the rotary table increment from A₀ to A₁, controlsthe tool Y-axis incremental motion from points 11 to 12. θ is thehalf-angle spanned by the chord line 12. From simple trigonometry, itfollows that, ##EQU1## This relationship can be used to relate toleranceT, as one determinant of groove finish, to the choice of rotary angleincrements A₀, A₁, . . . A_(i), . . . and Y axis movements Y₀, Y₁, . . .Y_(i). . . .

The relationship between the Y axis tool movement and the B axis tablerotation can now be established. The Y axis distance between points 11and 12 is

    Y.sub.1 =R.sub.G sin (2θ)                            Equation (2)

The corresponding angular movement of the rotary table from A₀ to A₁ is

    cos A.sub.1 =(-X.sub.1 /R.sub.B), where

    X.sub.1 =[point 13-point 10] or [point 24-point 20]

    X.sub.1 =R.sub.G cos (2θ), then ##EQU2##

To program the entire groove cutting cycle, a series of steps, eachhaving a unique Y and B axis relationship, must be specified. If eachstep is designated i=0,1,2, . . . N, with i=0 representing the initialtool and table positions at Y₀, A₀, the pair of corresponding Y and Baxis coordinates are

    Y.sub.i =R.sub.G sin (i*2θ) and                      Equation (4) ##EQU3## where ##EQU4## Each (Y.sub.i, A.sub.i) position pair is determined in sequence; the table is rotated and the tool moved continuously at depth D from coordinate to coordinate until a total of N movements, or discrete position targets, have been reached. N and θ are related such that N=180/θ. Once the choice of θ or T is made, a constant value of θ is used in equations (4) and (5). Thus, the B axis rotation A.sub.i is controlled by the Y axis movement Y.sub.i, as specified by the θ increments in the X-Y plane.

In the preferred embodiment, the Y axis movement is controlled byspecifying the B axis rotation increments A_(i). The sequence ofcoordinates can be determined by rewriting equation (4) and (5) suchthat ##EQU5##

    Y.sub.i =R.sub.G sin θ'                              Equation (7)

where ##EQU6## and solving equation (6) for X_(i) for use in equations(8) and (7).

After considerable investigation, it was determined that sequentialincrements of about 5° in B axis rotation, i.e., ΔA=A_(i) +₁ -A_(i) =5°,would give a sufficiently smooth finish in the O-ring groove to enablethe ring to be retained or trapped within the groove. For the four inchseat, the resulting groove finish was about 63. For seats having an I.D.in the range of about two to six inches, a satisfactory sequencing of Baxis rotation would be in the range of about three to seven degrees. ΔAis preferably an integer satisfying N=(180-2A₀)/ΔA. Larger values wouldnot produce a smooth enough groove to retain the O-ring, and smallervalues would be excessively time consuming.

The 5° increments in B axis rotation are translated by the computerthrough equations (6),(7) and (8) into the appropriate Y axisincrements. These are coupled together to form a simultaneous cuttingmotion that generates the groove.

It should be appreciated that conventional N.C. machines are notoperated according to the method set forth herein. Conventional controlsystems would interpret the simultaneous rotation of the B axis andvertical movement of the tool along the Y axis as a collision course andwould typically turn off or in some other way fail to execute. Ineffect, the present invention creates a novel combination of variablesfor controlling the N.C. machine.

The preferred embodiment of the inventive method described above wassuccessfully implemented on a numerically controlled K & T horizontalbore mill machine having a standard APT post processor provided byManufacturing Data and Systems, Inc., (MDSI), with the COMPACT-2 sourcecode, also available from MDSI. Since N.C. machines of this type havenot previously been used in the described manner, it was found necessaryto expand the data handling capability of the computerized controller.In particular, the X and Z axis registers of the control function withinthe link processor of the N.C. machine, may have to be modified tohandle up to ten significant digits.

Once the N.C. machine has been setup and loaded with the program, thegroove cutting operation preferably begins with a ball nose end mill torough out the groove in accordance with equation (6),(7) and (8). Then asecond pass is made with a flat bottomed end mill. As previouslydescribed, the end mill diameter of 0.156 inch cuts a groove of the samewidth to retain an O-ring having a 0.139 inch cross-section.

Referring now to FIGS. 10 through 12, a schematic representation ofvarious stages of groove cutting according to the invention are shown.FIG. 10 shows the tool 106 first penetrating the eccentric surface 52aat the 3 o'clock position corresponding to Y₀ and A₀. As the machiningprogresses, the tool moves downward along the Y axis and the fixturerotates along the B axis such that the tool 106 is positioned at 6o'clock in FIG. 11. As shown in FIG. 12, the tool 106 has now movedupward along the Y axis whereas the rotary table has continued rotatingsuch that the tool is at the 9 o'clock position on the eccentric surface52a. This is the position of maximum table rotation equal to (180°-2A₀).From this position the table reverses rotation, and the tool movesupward, then downward, eventually returning to the 3 o'clock positionshown in FIG. 10. The Y axis tool movement covers a total distance of4R_(G).

We claim:
 1. A method of producing a groove in a valve seat part, saidseat part having a hollow and generally cylindrical configuration abouta seat axis, and having at one end a generally saddle-shaped eccentricsurface adapted to conform to the contour of the inner surface of acylindrical valve main bore when the seat is transversely oriented tothe valve bore axis, said method comprising the steps of:mounting theseat part in a four-axis horizontal milling machine system such that thevalve seat part is affixed to a table adapted to rotate about a verticalaxis B, said part being oriented in the table fixture such that thecylindrical axis of the part is perpendicular to and intersects with theB axis, and the perpendicular distance from the B axis to every point onthe eccentric surfaace of the part is equal to the part radius R_(P) ;mounting an end mill tool on the machine such that the tool motion canbe specified by a coordinate system having a horizontal (X) axis, avertical (Y) axis, and a longitudinal (Z) axis, wherein the Z axiscoincides with the axis of the seat part, the Y axis is parallel to theB axis, and the X,Y,Z axes are mutually perpendicular; positioning thetool along the X axis until the tool contacts the valve seat eccentricsurface at the desired groove radius R_(G) ; rotating the part about theB axis until the tool is perpendicularly oriented relative to theeccentric surface; setting the tool motion along the Z axis for aconstant depth of mill cut; establishing a geometrical relationshipbetween the desired radius R_(G) of the groove and the radius R_(P) ofthe saddle-shaped curvature, for controlling the tool movement along theY axis and part rotation about the B axis, as the tool remains at afixed X axis locaation and constant Z axis groove milling depth; andoperating the machine system such that a generally rectangular groove ofconstant width and depth is milled into the eccentric surface of thepart, the groove sidewalls being substantially perpendicular to and thegroove base being parallel with the eccentric surface.
 2. The groovecutting method of claim 1 wherein the machine is controlled byspecifying a series of target coordinates of B axis rotation angle A_(i)and Y axis position Y_(i), such that the tool movement from coordinateto coordinate cuts a groove having a surface finish on the sidewalls andbase in the range of about 32-125.
 3. The groove cutting method of claim2 wherein the initial tool penetration of the eccentric surface is atcoordinates (A₀, Y₀), and the groove cutting cycle includes a totalvertical tool travel distance equal to 4R_(G), and maximum tablerotation angle equal to (180°-2A₀), before returning to coordinate (A₀,Y₀).
 4. The groove cutting method of claim 3 wherein the machineoperates in a first groove cutting cycle using a ball nose end mill torough out the groove, followed by a second cycle with a flat bottomedend mill to produce the generally rectangular groove.
 5. The groovecutting method of claim 3 wherein the machine is numerically controlledby a digital computer.
 6. The groove cutting method of claim 5 whereinthe coordinate pairs (Y_(i), A_(i)) are determined by the relationships,

    Y.sub.i =R.sub.G sin (i*2θ)

    A.sub.i =arc cos [-R.sub.G cos (i*2θ)/R.sub.P ]

where i is sequenced from 0 to N, and N equals 180/θ.
 7. The groovecutting method of claim 5 wherein the coordinate pairs (Y_(i), A_(i))are determined by the relationship

    A.sub.i =arc cos (-X.sub.i /R.sub.P)

    Y.sub.i =R.sub.G sin θ'

    θ'=arc cos (X.sub.i /R.sub.a)

where i is sequenced from 0 to N.
 8. The groove cutting method of claim7 wherein the sequential increment in rotation angle ΔA=A_(i+1) -A_(i),is in the range of about three to seven degrees.
 9. The groove cuttingmethod of claim 8 wherein the sequential increment of rotation angle,ΔA, is about 5°.